Incomplete perfect mendelsohn designs with block size 4 and holes of size 2 and 3

Let v , k , and n be positive integers. An incomplete perfect Mendelsohn design, denoted by k ‐IPMD( v , n ), is a triple ( X, Y , ) where X is a v ‐set (of points), Y is an n ‐subset of X , and is a collection of cyclically ordered k ‐subsets of X (called blocks ) such that every ordered pair ( a, b ) ∈ ( X × X)\(Y × Y ) appears t ‐apart in exactly one block of and no ordered pair ( a,b ) ∈ Y × Y appears in any block of for any t , where 1 ≤ t ≤ k − 1. In this article, the necessary conditions for the existence of a 4‐IPMD( v , n ), namely ( v − n ) ( v − 3n − 1 ) ≡ 0 (mod 4) and v ≥ 3n + 1, are shown to be sufficient for the case n = 3. For the case n = 2, these conditions are sufficient except for v = 7 and with the possible exception of v = 14,15,18,19,22,23,26,27,30. The latter result provides a useful application to the problem relating to the packing of perfect Mendelsohn designs with block size 4. © 1994 John Wiley & Sons, Inc.